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Mirrors > Home > MPE Home > Th. List > sspnv | Structured version Visualization version GIF version |
Description: A subspace is a normed complex vector space. (Contributed by NM, 27-Jan-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sspnv.h | ⊢ 𝐻 = (SubSp‘𝑈) |
Ref | Expression |
---|---|
sspnv | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑊 ∈ NrmCVec) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2824 | . . 3 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
2 | eqid 2824 | . . 3 ⊢ ( +𝑣 ‘𝑊) = ( +𝑣 ‘𝑊) | |
3 | eqid 2824 | . . 3 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈) | |
4 | eqid 2824 | . . 3 ⊢ ( ·𝑠OLD ‘𝑊) = ( ·𝑠OLD ‘𝑊) | |
5 | eqid 2824 | . . 3 ⊢ (normCV‘𝑈) = (normCV‘𝑈) | |
6 | eqid 2824 | . . 3 ⊢ (normCV‘𝑊) = (normCV‘𝑊) | |
7 | sspnv.h | . . 3 ⊢ 𝐻 = (SubSp‘𝑈) | |
8 | 1, 2, 3, 4, 5, 6, 7 | isssp 28504 | . 2 ⊢ (𝑈 ∈ NrmCVec → (𝑊 ∈ 𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (( +𝑣 ‘𝑊) ⊆ ( +𝑣 ‘𝑈) ∧ ( ·𝑠OLD ‘𝑊) ⊆ ( ·𝑠OLD ‘𝑈) ∧ (normCV‘𝑊) ⊆ (normCV‘𝑈))))) |
9 | 8 | simprbda 501 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑊 ∈ NrmCVec) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ⊆ wss 3939 ‘cfv 6358 NrmCVeccnv 28364 +𝑣 cpv 28365 ·𝑠OLD cns 28367 normCVcnmcv 28370 SubSpcss 28501 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-fo 6364 df-fv 6366 df-oprab 7163 df-1st 7692 df-2nd 7693 df-vc 28339 df-nv 28372 df-va 28375 df-sm 28377 df-nmcv 28380 df-ssp 28502 |
This theorem is referenced by: sspg 28508 ssps 28510 sspmlem 28512 sspmval 28513 sspz 28515 sspn 28516 sspimsval 28518 bnsscmcl 28648 minvecolem2 28655 hhshsslem1 29047 hhshsslem2 29048 |
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